Exponential growth kinetics

Thus, the Soai reaction is a template-directed self-replicating system that successfully maintains exponential growth kinetics and high autocatalytic efficiency over many turnovers. The results support the view that multiple and diverse ways exist to obtain chiral biomolecules via CPL or chiral inorganic crystals such as quartz combined with asymmetric autoctalysis. It is, however, important to remember that the Soai reaction must be carried out in nonaqueous solvents under prebiotically unrealistic conditions. 

First, asymmetric autocatalysis served to form chiral biomolecules. This reaction is a template-directed, self-replicating system, which successfully maintains ideal exponential growth kinetics and therefore high autocatalytic efficiency over many turnovers. 

Figure 22. (Bio)mesogens approaching pre-life states (7 a, 17, 18, 43] a) von Kiedrowski s and Orgel s minimal models of replication on the basis of self-complementary oligonucleotide DNA and RNA systems [44a-d, f-h] b) distant nucleic acid strand-analogs as matrix reaction models [7a, 18, 19, 39f-i] c) Rebek s self-replicational and evolutionary nucleoside analog model [45] d) von Kiedrowski s self-replicational amidinium-carboxylate model, being suggestive of exponential growth kinetics [44 e] e) Lehn s...

For those pesticides which are utilized as microbial growth substrates, sigmoidal rates of biodegradation are frequentiy observed (see Fig. 2). Sigmoidal data are more difficult to summarize than exponential (first-order) data because of their inherent nonlinearity. Sigmoidal rates of pesticide metabohsm can be described using microbial growth kinetics (Monod) however, four kinetics constants are required. Consequentiy, it is more difficult to predict the persistence of these pesticides in the environment. 

As already mentioned, the CO release exhibited an exponential growth to a steady-state level. The curves were fitted using a first-order kinetic scheme. Several sets of pyrolysis investigations were performed between 620 and 940 K. The lower temperature is a limit due to a large scatter in data received at lower temperatures and the upper temperature is limited by the temporal resolution of the concentration measurements. 

The solution to these kinetics is clearly unstable because it predicts unlimited exponential growth. Therefore, we need to assume that there is some sort of a death reaction. One form of this might be... 

Figure 5.17 Growth of a propene utilising Mycobacterium sp in batch culture exhibiting typical growth kinetics 1) lag phase 2) acceleration 3) exponential phase 4) deceleration 5) stationary phase 6) decline.

In a well mixed bioreactor a homogeneous suspension exists and typical growth kinetics an be observed as illustrated in Figure 5.17. Six phases can be distinguished the lag phase acceleration phase the exponential growth phase the deceleration phase the stationary phase and the phase at which death/decline occurs. 

KNIGHTS163 devised a procedure for determining the kinetic constants of a batch fermentation system which involves monitoring both the biomass and substrate concentrations, but without assuming a constant yield coefficient. Exponential growth in a batch fermenter is represented by equation 5.55 written as ... 

In self-replicating systems employing three starting constituents competition between constituents can occur [9.205]. Such processes are on the way to systems displaying information transfer, whereas the two-components ones are non-infor-mational. A shift from parabolic kinetics to exponential growth of the template concentration is required for a selection process to take place [9.197]. The evidence for self-replication on the basis of template-directed autocatalysis as in 184 requires detailed mechanistic investigation on the origin of the catalytic effects observed [9.206]. 

Table 6.1 shows the ACSL program and Figure 6.6 shows the results. Comparison of Figure 6.6 with Figure 6.1 shows that Monod kinetics can predict the cell growth from the start of the exponential growth phase to the stationary phase. 

As shown in Fig. 6B, a two-phase pattern occurred for the substrate uptake. It can be observed that during the exponential growth phase, sucrose assimilation by the bacteria was small, corresponding to about 20% of the initial amount introduced into the medium. However, after a 40-h process corresponding to the end of the growth phase, there was a rise in the substrate uptake, suggesting that the carbon source was directed to biosurfactant production, for the conditions tested. It should be emphasized that the fermentative process, when the medium was supplemented with microsalts and EDTA (Fig. 6A), generated a different substrate kinetics in comparison with that obtained for the nonsupplemented medium (Fig. 6B). 

Typical kinetic profiles (hybridoma). (A) Cell concentration and viability (B) glucose consumption (GLC) and lactate production (LAC) (C) monoclonal antibody production (mAb) (D) glutamine consumption (GLN) and ammonium production (NH4+) (E) specific growth rate (px) (F) alanine (ALA) and glycine (GLY) production. Adapted from Lee (2003). Symbols correspond to the experimental data and the lines to the manual curve fitting. Vertical lines indicate the instant at which exponential growth phase ended (gx < Px.max)-... 

Given that Eq. 6.1 (with D 2) applies to reaction-controlled flocculation kinetics, Eq. 6.54 implies that MM(t) [or MN(t)] must also exhibit an exponential growth with time. Therefore, by contrast with transport-controlled flocculation kinetics, a uniform value of the rate constant kmn cannot be introduced into the von Smoluchowski rate law, as in Eq. 6.17, to derive a mathematical model of the number density p,(t). Equations 6.22 and 6.24 indicate clearly that a uniform kinil leads to a linear time dependence in the... 

It is worthwhile remarking that a lot of these slow reactions do not proceed linearly and therefore they can only be modeled to a first approximation. The weathering of pyrite, e.g., will be catalyzed by microbes which are subject to exponential growth and death. These kinetics are taken into consideration in chapter 3.2.1.]... 

The flow velocities in flame systems are such that transport processes (diffusion and thermal conduction) make appreciable contributions to the overall flows, and must be considered in the analysis of the measured profiles. Indeed, these processes are responsible for the propagation of the flame into the fresh gas supporting it, and the exponential growth zone of the shock tube experiments is replaced by an initial stage of the reaction where active centres are supplied by diffusion from more reacted mixture sightly further downstream. The measured profiles are related to the kinetic reaction rates by means of the continuity equations governing the one-dimensional flowing system. Let Wi represent the concentration (g. cm" ) of any quantity i at distance y and time t, and let F,- represent the overall flux of the quantity (g. cm". sec ). Then continuity considerations require that the sum of the first distance derivative of the flux term and the first time derivative of the concentration term be equal to the mass chemical rate of formation q,- of the quantity, i.e. 

---